Exponential misuse
Dan Hoey
there was a listener's letter chiding the news writers for misusing
the term "exponential". Apparently they had used the term to describe
a rise in the United Nations workforce from 3000 to 5000. The
listener stated that this was an arithmetic increase, because an
exponential increase would have to be at least from a number to its
square: from 3000 to 9 million.
The colloquial misuse of mathematical terms (and "exponential" in
particular) is so widespread and egregious that it is disappointing to
hear such a blatantly spurious "correction". (For the benefit of the
confused, the mathematical term "exponential" refers to a function
that grows by the same multiplicative factor over every like interval.
Thus a workforce that increases by 67 per cent each time period, as
above, would indeed be exponential. A looser, but well-established,
technical usage includes functions that are in bounded proportion to
strict exponentials, such as a workforce that falls 1000 short of
doubling each time period, also consistent with the above data.) Of
course, the term has colloquially come to mean simply "rapidly
growing", so the entire issue is moot in the context of a broadcast to
a general audience.
Unfortunately, I didn't catch the details of the letter, so I'd
appreciate hearing from anyone who recalls the period of time over
which the UN workforce increased, or the exact numbers used (3000 and
5000 are my best recollection), or any other specifics of the
situation.
Please respond by e-mail to Ho...@AIC.NRL.Navy.Mil; I'll summarize if
there is interest. Thank you.
Dan Hoey
Ho...@AIC.NRL.Navy.Mil
Martin A. Mazur
ho...@aic.nrl.navy.mil (Dan Hoey) wrote:
>On National Public Radio's program "Morning Edition" for 27 September,
>there was a listener's letter chiding the news writers for misusing
>the term "exponential". Apparently they had used the term to describe
>a rise in the United Nations workforce from 3000 to 5000. The
>listener stated that this was an arithmetic increase, because an
>exponential increase would have to be at least from a number to its
>square: from 3000 to 9 million.
>
>The colloquial misuse of mathematical terms (and "exponential" in
>particular) is so widespread and egregious that it is disappointing to
>hear such a blatantly spurious "correction". (For the benefit of the
>confused, the mathematical term "exponential" refers to a function
>that grows by the same multiplicative factor over every like interval.
>Thus a workforce that increases by 67 per cent each time period, as
>above, would indeed be exponential. A looser, but well-established,
>technical usage includes functions that are in bounded proportion to
>strict exponentials, such as a workforce that falls 1000 short of
>doubling each time period, also consistent with the above data.) Of
>course, the term has colloquially come to mean simply "rapidly
>growing", so the entire issue is moot in the context of a broadcast to
>a general audience.
>
linear (i.e. arithmetic), polynomial, exponential (or just about anything
else) from only two data points. (Please direct flames on the use of "data
points" to the nearest dumpster). You'd need at least three points to even
_surmise_ that the growth is exponential, given that the data is probably
"noisy". One thing you can say about two measurements of a quantity is that
the quantity did (or, as in this case, did not) increase by an order of
magnitude in a period of time. But, with only two points, this says nothing
about the kind of growth that is occurring. The listeners correction, however,
is particulary egregious. If a quantity increases by squaring each time it is
measured, then (assuming equal time between measurements) its growth far
outstrips exponential growth.
--
Martin A. Mazur | 2nd Century thoughts on MTV:
The Applied Research Laboratory | "There is no public entertainment which
The Pennsylvania State University | does not inflict spiritual damage"
| - Tertullian
Bernard S. Greenberg
: In article <HOEY.95Se...@sun13.aic.nrl.navy.mil>,
: ho...@aic.nrl.navy.mil (Dan Hoey) wrote:
: >On National Public Radio's program "Morning Edition" for 27 September,
: >there was a listener's letter chiding the news writers for misusing
: >the term "exponential". Apparently they had used the term to describe
Beat ya. I sent them this this morning - I recommend you send them
a letter at the same address. I'm pissed at their stupidity.
From: Bernard S. Greenberg <b...@basistech.com>
To: mor...@npr.org
cc: b...@basistech.com, hu...@control.uchicago.edu
Subject: Exponential growth
Date: Wed, 27 Sep 1995 08:58:26 -0400
Darn it! Get somebody who knows something. Exponential growth does
NOT mean something "squares" then "cubes" or 3000 becomes 9000000.
Your "corrector" is more ignorant than that which s/he corrected.
Exponential growth is the same as geometric growth. It means that
some ratio is raised to an exponent - e. g.,
250, 2500, 25000, 250000 (where 10 is raised to a new power each time).
It does -not- mean that the original number is raised to a power. What
is more, if you only have two points ("10 years ago we had 40 members
and now we have 725"), if you don't know any of the intermediate points,
you can't tell ANYTHING about what kind of growth took place.
Exponential growth is usual when the rate of growth is proportional to how
many things are growing, as in a community of living, reproducing objects.
Please find somebody who -knows- something when technical issues are involved.
Don't take my word for it, ask an expert of your choosing.
Bernard S. Greenberg
b...@basistech.com
Ravi Sundaram
The best way to describe the term "exponential growth" to
non-technical people is to use the compound interest as the example.
When I heard the wrong correction on air, my wife was with me in the
car. She could immediately follow the example and understand the
difference.
Ravi Sundaram
09.28.95
Noel Gilmore
Sundaram) wrote:
Bless your heart and thank you (assuming, of course, that YOU are correct!).
Sincerely,
The Original Non-Technical Person,
Noel Gilmore
michael paine
>In article <HOEY.95Se...@sun13.aic.nrl.navy.mil>,
>>
>linear (i.e. arithmetic), polynomial, exponential (or just about anything
>else) from only two data points. (Please direct flames on the use of "data
>points" to the nearest dumpster). You'd need at least three points to even
>_surmise_ that the growth is exponential, given that the data is probably
>"noisy". One thing you can say about two measurements of a quantity is that
>the quantity did (or, as in this case, did not) increase by an order of
>magnitude in a period of time. But, with only two points, this says nothing
>about the kind of growth that is occurring. The listeners correction, however,
>is particulary egregious. If a quantity increases by squaring each time it is
>measured, then (assuming equal time between measurements) its growth far
>outstrips exponential growth.
I have another question which, perhaps, you can assist me with. What is meant by an increse in: "order of magnitude"? This is anothe=
r term that is often bandied about with, I think, little regard to its true meaning.
Thanks. Michael
Bernard S. Greenberg
: I have another question which, perhaps, you can assist me with. What is meant by an increse in: "order of magnitude"? This is anothe=
: r term that is often bandied about with, I think, little regard to its true meaning.
Easy: TEN.
Bernard S. Greenberg
b...@basistech.com
Martin A. Mazur
if you know what you are talking about.
The most common usage of the expression in the scientific community is that a
change of "an order of magnitude" means that a quantity has changed by
_roughly_ a _factor_ of ten. It is often used when talking in rough figures
about a quantity. Thus, if something increases by "an order of magnitude",
then you would expect that if you add a zero to the end of the original
quantity, you'd be in the ball park of the new value of the quantity.
Simililarly, if a quantity decreases be an order of magnitude, you'll be in
the right neighborhood if youmove the decimal point to the left one place. If
X was originally 784, then an increase to 8000 would be an increase of an
order of magnitude. Likewise, a decrease to 77 would mean a decrease of an
order of magnitude. An increase of three orders of magnitude means a
thousandfold increase.
There are other instances where the usage has a more general meaning. For
instance, it could mean that the exponent of the base under discussion (not
necessarily 10) has changed. Another example is from a branch of mathematics
called numerical analysis, which is used for solving complex problems via
algorithms that can be implemented on a computer, In this case, it is
desirable to know how the accuracy of an algorithm increases with a change in
the "step size" of the algorithm, i.e. "How much does the accuracy increase if
I tell the computer to step more finely in calculating the answer?" If the
step size is denoted by the letter "h", then to say that an algorithm "is
O(h^2)" (read "is Oh aitch squared") means that the accuracy increases as the
square of the step size used. The letter "O" stands for "order", or "order of
magnitude". In this example, the base is not 10, but an open parameter
that can be adjusted by the person using the algorithm. Thus, if an algorithm
is O(h^2) and the step size h is 0.1, the algorithm is accurate to roughly
0.01. Higher order algorithms are generally more desirable, but can be more
difficult to implement.
Rex Knepp
: I have another question which, perhaps, you can assist me with. What is meant by an increse in: "order of magnitude"? This is anothe=
: r term that is often bandied about with, I think, little regard to its true meaning.
To increase a number by "an order of magnitude" simply multiply it by
about ten. The new number will have an additional place to the left of
the decimal. That's what an order of magnitude means, at least in common
usage. For instance, My father's salary at my age was $8000 annually.
Mine is $50000. That means my salary is an order of magnitude higher
that my father's was 40 years ago. My sister earns $678000 per year;
her salary is an order of magnitude higher than mine, and two orders
higher that dad's was in 1955.
The *exact* numbers aren't important (or, in this example, true). It's
the power of ten. The phrase is used when one doesn't really want to
talk about exact, or even approximate numbers. If you say that the
speed of light is 150,000 miles/second, that's the right order of magnitude.
It's not very close, but it's a heck of a lot closer than 18,600 miles/second.
OK?
-30-
rex
Noel Gilmore
Knepp) wrote:
My sister earns $678000 per year;
> her salary is an order of magnitude higher than mine, and two orders
> higher that dad's was in 1955.
Would your sister like to adopt me?
>
> The *exact* numbers aren't important (or, in this example, true).
Oh, shucks!
Ruth Bygrave
[illuminating explanation of orders of magnitude snipped]
>The *exact* numbers aren't important (or, in this example, true). It's
>the power of ten.
I remember seeing a joke "translation" of various academic-ish usages,
which included the following line:
"Correct within an order of magnitude" "Wrong"
Regards, \/\/oof (please be patient, my newsreader isn't well)
Michael Callahan
David Casseres <cass...@apple.com> wrote:
>In article <44c9rj$1a...@hearst.cac.psu.edu>, m...@arlvax.arl.psu.edu
>(Martin A. Mazur) wrote:
>
>> I'd also like to add that there is no way of knowing whether the growth is
>> linear (i.e. arithmetic), polynomial, exponential (or just about anything
>> else) from only two data points.
I come into this discussion late, but...
I remember this letter claiming that the increase in UN staff wasn't
exponential because "the minimum exponential growth would be for the
number of staff to have squared"--sometime about a month ago, right?
I howled in agony at the thought that NO ONE at ME knew better, and
in fact tried to call them up to remonstrate with them (I was living
in Washington DC at the time and it was a local phonecall) but never
did it.
Did they ever broadcast a clarification?
Michael